Do Muslims worship the same God as Christians?

In one sense, yes. All men know there is one God because His power and divinity are revealed in creation. Romans 1 and 2 make that clear. Every man is bound to worship the Creator. But when a Muslim, after learning about Jesus Christ, deliberately rejects Him, he no longer worships the true God. Once you know that Christ is God and refuse to worship Him, your worship is false. The Catechism is incomplete on this because it does not make that distinction.